3.301 \(\int \frac{1}{\sqrt [4]{2+3 x^2} \left (4+3 x^2\right )} \, dx\)

Optimal. Leaf size=129 \[ -\frac{\tan ^{-1}\left (\frac{2 \sqrt [4]{2} \sqrt{3 x^2+2}+2\ 2^{3/4}}{2 \sqrt{3} x \sqrt [4]{3 x^2+2}}\right )}{2\ 2^{3/4} \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt{3 x^2+2}}{2 \sqrt{3} x \sqrt [4]{3 x^2+2}}\right )}{2\ 2^{3/4} \sqrt{3}} \]

[Out]

-ArcTan[(2*2^(3/4) + 2*2^(1/4)*Sqrt[2 + 3*x^2])/(2*Sqrt[3]*x*(2 + 3*x^2)^(1/4))]
/(2*2^(3/4)*Sqrt[3]) - ArcTanh[(2*2^(3/4) - 2*2^(1/4)*Sqrt[2 + 3*x^2])/(2*Sqrt[3
]*x*(2 + 3*x^2)^(1/4))]/(2*2^(3/4)*Sqrt[3])

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Rubi [A]  time = 0.0599293, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt{3 x^2+2}+2^{3/4}}{\sqrt{3} x \sqrt [4]{3 x^2+2}}\right )}{2\ 2^{3/4} \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{3 x^2+2}}{\sqrt{3} x \sqrt [4]{3 x^2+2}}\right )}{2\ 2^{3/4} \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[1/((2 + 3*x^2)^(1/4)*(4 + 3*x^2)),x]

[Out]

-ArcTan[(2^(3/4) + 2^(1/4)*Sqrt[2 + 3*x^2])/(Sqrt[3]*x*(2 + 3*x^2)^(1/4))]/(2*2^
(3/4)*Sqrt[3]) - ArcTanh[(2^(3/4) - 2^(1/4)*Sqrt[2 + 3*x^2])/(Sqrt[3]*x*(2 + 3*x
^2)^(1/4))]/(2*2^(3/4)*Sqrt[3])

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Rubi in Sympy [A]  time = 73.9714, size = 92, normalized size = 0.71 \[ \frac{\sqrt [4]{2} \sqrt{3} i \sqrt{- x^{2}} \Pi \left (- i; \operatorname{asin}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{3 x^{2} + 2}}{2} \right )}\middle | -1\right )}{6 x} - \frac{\sqrt [4]{2} \sqrt{3} i \sqrt{- x^{2}} \Pi \left (i; \operatorname{asin}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{3 x^{2} + 2}}{2} \right )}\middle | -1\right )}{6 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3*x**2+2)**(1/4)/(3*x**2+4),x)

[Out]

2**(1/4)*sqrt(3)*I*sqrt(-x**2)*elliptic_pi(-I, asin(2**(3/4)*(3*x**2 + 2)**(1/4)
/2), -1)/(6*x) - 2**(1/4)*sqrt(3)*I*sqrt(-x**2)*elliptic_pi(I, asin(2**(3/4)*(3*
x**2 + 2)**(1/4)/2), -1)/(6*x)

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Mathematica [C]  time = 0.173179, size = 135, normalized size = 1.05 \[ -\frac{4 x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{3 x^2}{2},-\frac{3 x^2}{4}\right )}{\sqrt [4]{3 x^2+2} \left (3 x^2+4\right ) \left (x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{3 x^2}{2},-\frac{3 x^2}{4}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{3 x^2}{2},-\frac{3 x^2}{4}\right )\right )-4 F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{3 x^2}{2},-\frac{3 x^2}{4}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((2 + 3*x^2)^(1/4)*(4 + 3*x^2)),x]

[Out]

(-4*x*AppellF1[1/2, 1/4, 1, 3/2, (-3*x^2)/2, (-3*x^2)/4])/((2 + 3*x^2)^(1/4)*(4
+ 3*x^2)*(-4*AppellF1[1/2, 1/4, 1, 3/2, (-3*x^2)/2, (-3*x^2)/4] + x^2*(2*AppellF
1[3/2, 1/4, 2, 5/2, (-3*x^2)/2, (-3*x^2)/4] + AppellF1[3/2, 5/4, 1, 5/2, (-3*x^2
)/2, (-3*x^2)/4])))

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{\frac{1}{3\,{x}^{2}+4}{\frac{1}{\sqrt [4]{3\,{x}^{2}+2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3*x^2+2)^(1/4)/(3*x^2+4),x)

[Out]

int(1/(3*x^2+2)^(1/4)/(3*x^2+4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, x^{2} + 4\right )}{\left (3 \, x^{2} + 2\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((3*x^2 + 4)*(3*x^2 + 2)^(1/4)),x, algorithm="maxima")

[Out]

integrate(1/((3*x^2 + 4)*(3*x^2 + 2)^(1/4)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((3*x^2 + 4)*(3*x^2 + 2)^(1/4)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [4]{3 x^{2} + 2} \left (3 x^{2} + 4\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(3*x**2+2)**(1/4)/(3*x**2+4),x)

[Out]

Integral(1/((3*x**2 + 2)**(1/4)*(3*x**2 + 4)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, x^{2} + 4\right )}{\left (3 \, x^{2} + 2\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((3*x^2 + 4)*(3*x^2 + 2)^(1/4)),x, algorithm="giac")

[Out]

integrate(1/((3*x^2 + 4)*(3*x^2 + 2)^(1/4)), x)